Constructive Reverse Mathematics

Diener, Hannes

doi:10.48550/arxiv.1804.05495

Cited by 4 publications

(12 citation statements)

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“…Constructively, one can not prove that N N and B N are in bijection. However, KT is equivalent to the existence of a continuous bijection B N → N N with a continuous modulus of continuity [11]. Furthermore, KT yields a continuous bijection…”

Section: Axioms On Treesmentioning

confidence: 99%

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Church's thesis and related axioms in Coq's type theory

Forster

2020

Preprint

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Church's thesis" (CT) as an axiom in constructive logic states that every total function of type N → N is computable, i.e. definable in a model of computation. CT is inconsistent in both classical mathematics and in Brouwer's intuitionism since it contradicts Weak König's Lemma and the fan theorem, respectively. Recently, CT was proved consistent for (univalent) constructive type theory.Since neither Weak König's Lemma nor the fan theorem are a consequence of just logical axioms or just choice-like axioms assumed in constructive logic, it seems likely that CT is only inconsistent with a combination of classical logic and choice axioms. We study consequences of CT and its relation to several classes of axioms in Coq's type theory, a constructive type theory with a universe of propositions which does neither prove classical logical axioms nor strong choice axioms.We thereby provide a partial answer to the question which axioms may preserve computational intuitions inherent to type theory, and which certainly do not. The paper can also be read as a broad survey of axioms in type theory, with all results mechanised in the Coq proof assistant.

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Section: Axioms On Treesmentioning

confidence: 99%

“…definable in a model of computation. CT is well-studied as part of Russian constructivism [32] and in the field of constructive reverse mathematics [24,11].…”

Section: Introductionmentioning

confidence: 99%

Church's thesis and related axioms in Coq's type theory

Forster

2020

Preprint

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“…A similar analysis is carried out in publications on constructive reverse mathematics, where one does not assume classical logic, but the axiom of countable or even dependent choice. However, we are only aware of an analysis for the compactness theorem for propositional logic rather than first-order logic, which is equivalent to WKL for decidable trees [12].…”

Section: Compactness and Weak Kőnig's Lemmamentioning

confidence: 99%

“…Note that in the context of constructive reverse mathematics (e.g. in [12]) WKL is only stated for decidable trees. We however need both notions and thus distinguish them by an index.…”

Section: Compactness and Weak Kőnig's Lemmamentioning

confidence: 99%

“…WKL is a consequence of compactness for decidable models. The proof is essentially the same as the one for propositional logic and WKL D by Diener [12] and the one for first-order logic using the classical base theory RCA 0 by Simpson [65]. Intuitively, given a tree τ , one can construct a formula ϕ n over the siganture Σ N which is satisfiable iff τ contains an element of length n. Definition 32.…”

Section: Compactness and Weak Kőnig's Lemmamentioning

confidence: 99%

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Completeness Theorems for First-Order Logic Analysed in Constructive Type Theory

Forster

Kirst

Wehr

2019

Lecture Notes in Computer Science

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We study various formulations of the completeness of firstorder logic phrased in constructive type theory and mechanised in the Coq proof assistant. Specifically, we examine the completeness of variants of classical and intuitionistic natural deduction and sequent calculi with respect to model-theoretic, algebraic, and game-theoretic semantics. As completeness with respect to the standard model-theoretic semantics à la Tarski and Kripke is not readily constructive, we analyse connections of completeness theorems to Markov's Principle and Weak Kőnig's Lemma and discuss non-standard semantics admitting assumption-free completeness. We contribute a reusable Coq library for first-order logic containing all results covered in this paper. 1 Accepted in Russian constructivism while in conflict with Brouwer's intuitionism arXiv:2006.04399v1 [cs.LO] 8 Jun 2020 Fact 6. T and F are data types and Γ ϕ and Γ c ϕ are enumerable. Proof. By the techniques discussed in [16], e.g. Fact 3.19. The standard model-theoretic completeness proofs analysed in Section 3 require the assumption of Markov's Principle. A proposition P : P is called stable if ¬¬P → P and, analogously, a predicate p : X → P is called stable if p x is stable for all x. A synthetic version of Markov's Principle states that satisfiability of Boolean sequences is stable (cf. [51]): MP := ∀f : N → B. ¬¬(∃n. f n = tt) → ∃n. f n = tt Note that MP is trivially implied by Excluded Middle EM := ∀P : P. P ∨ ¬P . Moreover, MP regulates the behaviour of computationally tractable predicates: Fact 7. MP holds iff all enumerable predicates on discrete types are stable. Proof. The direction from left to right is Fact 2.18 in [16]. For the reverse direction assume that enumerable predicates on discrete types are stable. Let f : N → B and let p : 1 → P be defined by p x := ∃n. f n = tt. The predicate p is enumerable by f n := if f n then else ∅. Stability of p is now equivalent to ¬¬(∃n. f n = tt) → (∃n. f n = tt).As a consequence of Fact 6 and Fact 7, MP implies that the deduction systems Γ ϕ and Γ c ϕ are stable. In fact, only these stabilities are required for the standard model-theoretic completeness proofs discussed in the next section and they are equivalent to MP L , a version of Markov's Principle stated for the call-by-value λ-calculus L [57,22] and its halting problem E: ¬¬Es → EsWe will prove the following in Section 3.5: Lemma 8. MP L , stability of Γ ϕand stability of Γ c ϕ are all equivalent. Completeness Theorems for FOL Analysed in Constructive Type Theory 73 Model-Theoretic SemanticsThe first variant of semantics we consider is based on the idea of interpreting terms as objects in a model and embedding the logical connectives into the metalogic. A formula is considered valid if it is satisfied by all models. The simplest case is Tarski semantics, coinciding with classical deduction via Henkin's completeness proof factoring through a (constructive) model-existence theorem [30]. Kripke semantics, coinciding with intuitionistic deduction, add more structure by connec...

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Algebras of Complemented Subsets

Petrakis¹,

Misselbeck-Wessel²

2022

Revolutions and Revelations in Computability

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We introduce cs-topologies, or topologies of complemented subsets, as a new approach to constructive topology that preserves the duality between open and closed subsets of classical topology. Complemented subsets were used successfully by Bishop in his constructive formulation of the Daniell approach to measure and integration. Here we use complemented subsets in topology, in order to describe simulataneously an open set, the first-component of an open complemented subset, and its complement as a closed set, the second component of an open complemented subset. We analyse the canonical cs-topology induced by a metric, and we introduce the notion of a mudulus of openness for a cs-open subset of a metric space. Pointwise and uniform continuity of functions between metric spaces are formulated with respect to the way these functions inverse open complemented subsets together with their moduli of openness. The addition of moduli of openness in the concept of an open subset, given a base for the cs-topology, makes possible to define the notion of uniform continuity of functions between csb-spaces, that is cs-spaces with a given base. In this way, the notions of pointwise and uniform continuity of functions between metric spaces are directly generalised to the notions of pointwise and uniform continuity between csb-topological spaces. Constructions and facts from the classical theory of topologies of open subsets are translated to our constructive theory of topologies of open complemented subsets.

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Constructive Reverse Mathematics

Cited by 4 publications

References 40 publications

Church's thesis and related axioms in Coq's type theory

Church's thesis and related axioms in Coq's type theory

Completeness Theorems for First-Order Logic Analysed in Constructive Type Theory

Algebras of Complemented Subsets

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